WELL-POSEDNESS AND BLOW-UP PHENOMENA FOR A GENERALIZED CAMASSA-HOLM EQUATION

. We ﬁrst establish the local existence and uniqueness of strong solutions for the Cauchy problem of a generalized Camassa-Holm equation in nonhomogeneous Besov spaces by using the Littlewood-Paley theory. Then, we prove that the solution depends continuously on the initial data in the cor- responding Besov space. Finally, we derive a blow-up criterion and present a blow-up result and a blow-up rate of the blow-up solutions to the equation.


1.
Introduction. Recently, Novikov in [39] proposed the following integrable quasilinear scaler evolution equation of order 2: where = 0 is a real constant. He showed [39] that (1.1) possesses a hierarchy of local higher symmetries. Eq.(1.1) belongs to the following class [39]: (1 − ∂ 2 x )u t = F (u, u x , u xx , u xxx ), (1.2) which has attracted much attention on the possible integrable members of (1.2). The first well-known integrable member of (1.2) is the Camassa-Holm (CH) equation [4] (1 − ∂ 2 x )u t = −(3uu x − 2u x u xx − uu xxx ). The CH equation can be regarded as a shallow water wave equation [4,5,18]. It is completely integrable [4,8,19,16], has a bi-Hamiltonian structure [7,28], and admits exact peaked solitons of the form ce −|x−ct| , c > 0, which are orbitally stable [20]. It is worth mentioning that the peaked solitons present the characteristic for 5494 JINLU LI AND ZHAOYANG YIN the traveling water waves of greatest height and largest amplitude and arise as solutions to the free-boundary problem for incompressible Euler equations over a flat bed, cf. [10,14,15,40]. The local well-posedness for the Cauchy problem of the CH equation in Sobolev spaces and Besov spaces was discussed in [11,12,21]. It was shown that there exist global strong solutions to the CH equation [9,11,12] and finite time blow-up strong solutions to the CH equation [9,11,12,13]. The global conservative and dissipative solutions of CH equation were discussed in [2,3].
The second well-known integrable member of (1.2) is the Degasperis-Procesi (DP) equation [24]: . The DP equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the CH shallow water equation [25]. The DP equation is integrable and has a bi-Hamiltonian structure [23]. An inverse scattering approach for computing n-peakon solutions to the DP equation was presented in [38,17]. Its traveling wave solutions were investigated in [33,41]. The local well-posedness of the Cauchy problem of the DP equation in Sobolev spaces and Besov spaces were established in [29,30,47]. Similar to the CH equation, the DP equation has also global strong solutions [35,48,50] and finite time blowup solutions [26,27,35,36,47,48,49,50]. On the other hand, it has global weak solutions [6,26,49,50]. Although the DP equation is similar to the CH equation in several aspects, these two equations are truly different. One of the novel features of the DP different from the CH equation is that it has not only peakon solutions [23] and periodic peakon solutions [49], but also shock peakons [37] and the periodic shock waves [27].
The third well-known integrable member of (1.2) is the Novikov equation [31] (1 − ∂ 2 x )u t = 3uu x u xx + u 2 u xxx − 4u 2 u x . The most difference between the Novikov equation and the CH and DP equations is that the former one has cubic nonlinearity and the latter ones have quadratic nonlinearity. It was showed that the Novikov equation is integrable, possesses a bi-Hamiltonian structure, and admits exact peakon solutions u(t, x) = ± √ ce |x−ct| , c > 0 [31]. The local well-posedness for the Novikov equation in Sobolev spaces and Besov spaces was studied in [43,44,45,46]. The global existence of strong solutions was established in [43] under some sign conditions and the blow-up phenomena of the strong solutions were shown in [46]. The global weak solutions for the Novikov equation were discussed in [42].
To our best knowledge, the Cauchy problem of Eq.(1.1) has not been studied yet. In this paper, we mainly study the Cauchy problem of Eq.(1.1). Since Eq.(1.1) has the similar structure with the Camassa-Holm equation, we call it as a generalized Camassa-Holm equation. Letting v(t, x) = u( 2 t, x), then one can transform the Cauchy problem of Eq.(1.1) into the following equivalent form: (1.4) In this paper, using the Littlewood-Paley theory, we establish the local existence and uniqueness of solutions to (1.4) in nonhomogeneous Besov spaces. For the stability of the solution, lots of papers just proved it in lower regularity Besov spaces B s p,r , s < s when the initial data u 0 ∈ B s p,r . We use a priori estimate of solutions to transport equation in Besov spaces B 1+ 1 p p,r [34] and apply the method introduced by Danchin [22] to get the continuity with respect to initial data. By the structure of the equation (1.4) and Gronwall's inequality, we present a blow-up result provided the initial data m 0 satisfies some conditions and give a blow-up rate for the blow-up solutions of (1.4).
Our paper is organized as follows. In Section 2, we give some preliminaries which will be used in the sequel. In Section 3, we establish the local existence and uniqueness of solutions of the Cauchy problem (1.4) in Besov spaces. In Section 4, we show the continuity of solutions to (1.4) with respect to initial data. In Section 5, we present a blow-up result to (1.4) and give a blow-up rate for the blow-up solutions of (1.4).
Notation. In the following, we denote by * the convolution and by ·, · the action between S (R) and S(R). Given a Banach space X, we denote its norm by · X .
2. Preliminaries. In this section, we will recall some facts on the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces and their some useful properties. We will also recall the transport equation theory, which will be used in our work. For more details, the readers can refer to [1,21].
Then for all u ∈ S (R d ), we can define the nonhomogeneous dyadic blocks as follows. Let ∆ q u 0, if q ≤ −2, where the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of u.
Remark 2.2. [1] (1) The low frequency cut-off operator S q is defined by (2) The Littlewood-Paley decomposition is quasi-orthogonal in L 2 in the following sense: for all u, v ∈ S (R d ).
(3) Thanks to Young's inequality, we get where C is a positive constant independent of p and q.
In the following lemma, we list some important properties of Besov spaces.
is an algebra, provided that s > d p or s = d p and r = 1.
Remark 2.6. We will see that, the product estimate (2.2) is one of the keys to our work. We mention here the simpler case, say, if 1 Now we state some useful results in the transport equation theory, which are crucial to the proofs of our main theorems later.
) solves the following 1-D linear transport equation: then there exists a constant C depending only on p, r and σ, such that the following statements hold: ) and the inequalities of Lemma 2.8 can hold true. Moreover, if r < ∞, then f ∈ C([0, T ]; B σ p,r (R)).
. We introduce a lemma to give a priori estimate for the solutions of 1-D linear ) solves the 1-D linear transport equation (T ), then there exists a constant C depending only on p, r and σ, such that the following statement holds: In the following, since all spaces of functions are over R, for simplicity, we drop R in our notations of function spaces if there is no ambiguity.
3. Local existence and uniqueness. In the section, we will establish the local existence and uniqueness of solutions for the Cauchy problem (1.4) in Besov spaces. Since m = u − u xx , we can rewrite (1.4) as follows: (3.1) We now give a definition and show the local existence and uniqueness result.
Theorem 3.2. Let 1 ≤ p, r ≤ ∞, s > 1 + 1 p , (or s = 1 + 1 p , r = 1, 1 ≤ p < ∞) and m 0 ∈ B s p,r . There exists a time T > 0 such that (3.1) has a unique solution m ∈ E s p,r (T ). We use four steps to prove the local existence and one proposition to prove the uniqueness of the solution to (3.1).

First
Step. constructing approximate solutions. Starting from m 0 0, we define by induction a sequence (m n ) n∈N of smooth functions by solving the following linear system: Since m 0 ∈ B s p,r , then all initial data S n+1 m 0 ∈ B ∞ p,r and ||S n+1 m 0 || B s p,r ≤ C||m 0 || B s p,r . Apply Lemma 2.9 and by induction, for every n ≥ 1, T n−1 has a unique solution m n in C 1 ([0, T ]; B ∞ p,r ). Obviously, m n belongs to E s p,r (T ) for all positive T .

Second
Step. uniform bounds. Using the embedding relations, product laws (Lemmas 2.4 − 2.5 ) and Lemma 2.8, we see that for all n ∈ N, where U n (t) t 0 ||m n (τ )|| B s p,r dτ and C ≥ 1. We now fix a T > 0 such that C 2 ||m 0 || B s p,r T < 1. By induction, we gain In fact, suppose it is valid for n, by (3.2) and (3.3), we obtain . This entails that (u n xx +u n x )m n+1 x is bounded in L ∞ ([0, T ]; B s−1 p,r ). Then we can conclude that (m n ) n∈N is bounded in E s p,r (T ). Third Step. convergence. We will prove that: (m n ) n∈N is a Cauchy sequence of C([0, T ]; B s−1 p,r ). Indeed, for all n ∈ N, we have From the definition of operator S n , we get

Thanks to Remark 2.2 and Definition 2.3, we readily have
. Arguing by induction with respect to the index n, we can obtain Therefore, we deduce that (m n ) n∈N is a Cauchy sequence in C([0, T ]; B s−1 p,r ) and converges to a limit function m ∈ C([0, T ]; B s−1 p,r ). Fourth Step. conclusion. We will show that m belongs to E s p,r (T ) and satisfies (3.1). Since (m n ) n∈N is bounded in L ∞ (0, T ; B s p,r ), the Fatou property for Besov spaces guarantees that m also belongs to L ∞ (0, T ; B s p,r ). Now, as (m n ) n∈N converges to m ∈ C([0, T ]; B s−1 p,r ), an interpolation argument (in Lemma 2.4) ensures that the convergence actually holds true in C([0, T ]; B s p,r ) for any s < s. It is then easy to pass to the limit in (T n ) and to conclude that m(t, x) is a solution to (3.1). Indeed, for any test function ϕ ∈ C 1 ([0, T ]; S), ∀t ∈ [0, T ] , besides the easier terms, the most complicated terms in the convergence of the approximation are Finally, because m belongs to L ∞ (0, T ; B s p,r ), in view of Lemma 2.9, the solution m belongs to C([0, T ]; B s p,r ) (resp., C w ([0, T ]; B s p,r )) if r < ∞ (resp., r = ∞). Applying Lemma 2.11, we see that ∂ t u is in C([0, T ]; B s−1 p,r ) if r is finite, and in L ∞ ([0, T ]; B s−1 p,r ) otherwise. Hence we conclude that the solution m ∈ E s p,r (T ). Now, we turn to the uniqueness of the solution to (3.1). In fact, it is a straightforward corollary of the following proposition.

4.
Continuity with respect to initial data. In this section, we will prove the solution of (1.4) guaranteed by Theorem 3.2 depends continuously on the initial data. First, we introduce a useful lemma in the proof of the next theorem.
Proof. By Proposition 3.3, we get ||m n − m ∞ || L ∞ (0,T ;B s−1 p,r ) tends to zero as n → ∞. So we only need to prove ||∂ x m n − ∂ x m ∞ || L ∞ (0,T ;B s−1 p,r ) tends to zero as n → ∞ where r is finite. According to Theorem 3.2, we can find M > 0 such that for all n ∈ N, Denote v n = ∂ x m n . Note that v n solves the following linear transport equation: Applying Lemmas 2.8 − 2.10, product laws and embedding relations, we get Since the sequence (a n ) n∈N is uniformly bounded in C([0, T ]; B s p,r ) and tends to a ∞ in C([0, T ]; B s−1 p,r ), Lemma 4.1 tells us that w n tends to v ∞ = ∂ x m ∞ in C([0, T ]; B s−1 p,r ). Let > 0. Since m n tends to m ∞ in C([0, T ]; B s−1 p,r ), the last term is less than 1 2 for large n. Combining the above convergence results and Lemma 2.8, one concludes that for large enough n ∈ N, Hence, thanks to Gronwall's inequality, we get , for some constantC depending only on s, p, r M and T . This complets the proof in the case r < ∞.
When tends to zero as j → ∞ and ||m n −m ∞ || L ∞ (0,T ;B s−1 p,r ) tends to zero as n → ∞. Then (4.1) can be made arbitrarily small for j large enough. For fixed j, we then let n tend to infinity so that (4.2) tends to zero, and we conclude that m n (t) − m ∞ (t), φ tends to zero. Then this completes the proof in the case r = ∞.
Applying Λ s to both sides of (1.4) and taking the L 2 inner product with Λ s m, we have For the last term of the right-hand side of (5.2), applying Lemma 2.10, we obtain The similar argument allows us to show that which implies that

JINLU LI AND ZHAOYANG YIN
Now, let us consider the following initial value problem By (1.4), one easily has m(t, q(t, x)) = m 0 , which leads to ||m(t)|| L ∞ = ||m 0 || L ∞ . Let p = 1 2 e −|x| . As u = p * m and u x = ∂ x p * m, then we have Using Gronwall's inequality and (5.6), we get This completes the proof of Lemma 5.1. Proof. On the one hand, by Lemma 5.1 and Sobolev's imbedding theorem, it is clear that if the slope of the solution tends to minus infinity in finite time, then T < ∞.
On the other hand, we first multiply (1.4) by m and integrate by parts to get Then, differentiating (1.4) with respect to the spatial variable x, multiplying the obtained equation by m x and integrating by parts, we obtain Next, differentiating (1.4) twice with respect to the spatial variable x, multiplying the obtained equation by m xx and integrating by parts, we have Combining (5.9), (5.10) and (5.11), we get If m x is bounded from below on [0, T ) × R, i.e., if there exists M > 0 such that Applying Gronwall's inequality and Lemma 5.1, we conclude that ||u(t)|| H s is uniformly bounded in t ∈ [0, T ) which contradicts the fact that the solution blows up in finite time T . This completes the proof of Theorem 5.2. Proof. Take K = 3||m 0 || L ∞ . From (1.4) and (5.8), we see that which associating with (5.7) yields ∂ t m x (t, q(t, x)) ≤ −m 2 x (t, q(t, x)) + K 2 . (5.13) Set y(t) = m x (t, q(t, x 0 )). According to the assumption that m x (x 0 ) < −K, the inequality (5.13) at x 0 becomes y (t) ≤ −y 2 (t) + K 2 . Then we claim: y(t) < −K.